In [1]} Peterzil and Pillay showed that, in an o-minimal structure with definable choice functions, if a definable family of closed and bounded sets is consistent (i.e. has the finite intersection property), then it has a finite transversal (i.e. there exists a finite set that intersects each set in the family). They derived this from the work of Dolich [2]} on forking in o-minimal theories –in particular Dolich's proof that, in an o-minimal expansion of a fieldMore specifically, Dolich works in the slightly more general setting of a “nice" o-minimal theory., a formula \(\varphi (x,b)\)  does not fork (or does not divide) over a set \(A\)  if and only if it is “good over \(A\) ". It is easy to see that being good over \(A\)  implies that \(\varphi (x,b)\)  extends to a (global) \(A\) -definable type. This equivalence between formulas not forking over a set \(A\)  and extending to \(A\) -definable types was conjectured for all dp-minimal theories by Simon [3]}, in the case where \(A\)  is a model. The most general result up to date in this direction is due to Simon and Starchenko [4]}, who proved that the conjecture holds for a large class of dp-minimal theories (see Theorem REF ), including those that are linearly ordered (e.g. o-minimal) or have definable Skolem functions (e.g. the theory of the p-adics). The aforementioned result in [1]} highlights how this line of research on forking in tame theories can produce results about intersecting definable families of closed sets in some definable topology with some appropriate notion of topological compactness.
